Heisenberg uniqueness pairs for the hyperbola

2019/09/26 by Giri, Deb Kumar, Rawat, Rama
#42B10 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.12076

Abstract

Let Γ be the hyperbola \(x,y)∈\mathbb R² : xy=1\ and Λ_β be the lattice-cross defined by Λ_β=(\mathbb Z×\0\)∪(\0\×β\mathbb Z) in \mathbb R², where β is a positive real. A result of Hedenmalm and Montes-Rodríguez says that (Γ,Λ_β) is a Heisenberg uniqueness pair if and only if β≤1. In this paper, we show that for a rational perturbation of Λ_β, namely Λ_β^θ=((\mathbb Z+\θ\)×\0\)∪(\0\×β\mathbb Z), where θ=1/p,~for some~p∈\mathbb N and β is a positive real, the pair (Γ,Λ_β^θ) is a Heisenberg uniqueness pair if and only if β≤p.

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